# Symmetry – Definition, Examples, Practice Problems, FAQs

Created on Jan 10, 2024

Updated on January 10, 2024

Welcome to another enlightening journey with Brighterly, where learning is not only about acquiring knowledge, but also about kindling curiosity and fostering a lifelong love for exploration. In our quest to make mathematics a fascinating adventure for young minds, we delve into an intriguing concept today – Symmetry.

Symmetry, a concept that often evokes images of perfectly balanced shapes and patterns, is not just an abstract mathematical idea. It is a pervasive principle that underscores the structure of our world, making it comprehensible and aesthetically pleasing. But what exactly is symmetry? How do we recognize it in different shapes and patterns? And how does it manifest in our everyday lives?

## Line of Symmetry

The term line of symmetry, often known as a mirror line or reflection line, is a line that divides a figure into two congruent halves. It’s an invisible axis that gives the figure balance and allows it to reflect an identical image on either side. A shape or design is said to have symmetry if you can draw a line through it and the figure appears the same on either side of the line. It’s akin to looking into a mirror, seeing an identical reflection of oneself. Shapes with a line of symmetry are integral in both the natural and artistic world, providing a sense of harmony and balance that is universally pleasing to the eye.

## Vertical Line of Symmetry

The concept of a vertical line of symmetry comes into play when the line of symmetry runs straight from the top to the bottom, dividing the figure into two congruent halves. The vertical line is one that moves straight up and down along the y-axis. In geometric shapes, the vertical line of symmetry can be observed in many instances, such as rectangles, squares, ovals, or even the letter ‘A’. The symmetry adds to their aesthetic appeal and structural strength. It’s fascinating to note how these symmetric structures are prevalent around us, both in man-made structures and in nature.

## Horizontal Line of Symmetry

In contrast to a vertical line of symmetry, a horizontal line of symmetry runs left to right, straight across the x-axis. This type of symmetry can cut a figure in half, horizontally forming two mirror images above and below the line. A classic example is the English letter ‘B’ or a perfect oval. The horizontal line of symmetry is also a common feature in several mathematical formulas and theorems. In nature, we can see horizontal lines of symmetry in animals and plants, highlighting the role of symmetry in biological survival and evolution.

## Diagonal Line of Symmetry

The diagonal line of symmetry is an intriguing aspect of symmetry that goes beyond the vertical and horizontal divides. This type of symmetry occurs when a line runs diagonally, splitting the figure into two identical halves. Not all shapes possess diagonal symmetry, but those that do, like a square or some types of rhombus, showcase an additional layer of balance and harmony. It’s amazing to see how the diagonal line of symmetry plays a role in artistic creations, such as patterned tiles and intricate carpet designs, embodying beauty and consistency.

## One Line of Symmetry

When a figure has one line of symmetry, it means that it can only be divided into two identical halves in one way. This can often be seen in various objects, including letters, numbers, and geometric shapes. For instance, the letter ‘A’ in the English alphabet, when folded along its vertical line of symmetry, forms two identical halves. This kind of symmetry can also be observed in isosceles triangles, where a line drawn from the vertex angle to the base forms two equal halves. Understanding single-line symmetry is a fundamental stepping stone in the study of more complex symmetrical structures, giving us an initial sense of balance and proportion in design and nature.

## Two Lines of Symmetry

When a figure has two lines of symmetry, it means that the figure can be divided into identical halves twice along two different lines. One of the simplest examples is the English letter ‘H’, which can be split evenly along both a vertical and a horizontal axis. Another prime example is an equilateral triangle, which has three lines of symmetry. Understanding multiple lines of symmetry deepens our grasp of symmetry, leading us to appreciate the intricate balance and proportion in more complex designs and structures.

## Lines of Symmetry for Different Shapes

Different shapes possess varying numbers of lines of symmetry. For instance, a square has four lines of symmetry – two vertical and two diagonal – while a rectangle only has two, one vertical and one horizontal. Circles are unique in that they have an infinite number of lines of symmetry, as they can be divided along any diameter. By recognizing lines of symmetry in different shapes, we develop an intuitive understanding of geometric concepts and a sharper eye for symmetrical patterns in the world around us.

## Types of Symmetry

Symmetry isn’t just about mirror images. There are several other types of symmetry which include translation, rotational, reflexive, and glide symmetry. These forms of symmetry go beyond simple reflection and incorporate concepts of movement and rotation. Understanding these different types of symmetry helps us appreciate the diversity and complexity of patterns in art, nature, and mathematics.

## Translation Symmetry

Translation symmetry refers to the situation where a figure can be moved, or “translated,” without rotation or reflection and still superpose exactly onto the original figure. This type of symmetry is commonly seen in patterns or tessellations. Wallpaper designs, crystal structures, and honeycombs are examples of translation symmetry, where a pattern is repeated at regular intervals. It serves as an example of how symmetry can incorporate elements of movement and change while maintaining a consistent pattern.

To consolidate your knowledge, we recommend using our Symmetry worksheets from Brighterly.

## Rotational Symmetry

With rotational symmetry, a figure can be rotated around a central point by less than a full circle (360 degrees) and still resemble the original figure. A classic example is a snowflake, which typically exhibits six-fold rotational symmetry, looking the same after a rotation of 60 degrees. Rotational symmetry is a central concept in many fields, including art, architecture, biology, and advanced areas of mathematics.

## Reflexive Symmetry

Reflexive symmetry, also known as reflexion or reflection symmetry, refers to a shape that can be reflected over a line (the line of symmetry) and still resemble its original shape. This is perhaps the most intuitively understood type of symmetry, observed when one half of a figure is a mirror image of the other half. A classic example is a heart shape or the wings of a butterfly.

## Glide Symmetry

Glide symmetry, also known as glide reflection, is a type of symmetry where a figure is reflected and translated (or glided) along a line, resulting in an identical image. This type of symmetry combines reflection and translation, where the figure is first mirrored and then shifted along the direction of the line. The footprints of a person walking in a straight line represent an example of glide symmetry.

## Symmetrical Shapes

Symmetrical shapes are fundamental to understanding symmetry. These are shapes that can be split evenly in at least one way, forming two or more parts that mirror each other. These include polygons such as squares, rectangles, and circles, as well as more complex shapes like hexagons, octagons, and so on. Recognizing symmetrical shapes enhances our ability to visualize and comprehend geometrical concepts and to appreciate the beauty and harmony in balanced designs.

## Symmetry in Real Life

Symmetry in real life is ubiquitous. From the wings of a butterfly, to architectural structures, to human faces, symmetry is a fundamental characteristic of both natural and man-made objects. Its wide-ranging presence illustrates the importance of symmetry not only as an aesthetic quality but also as a principle underlying the fundamental laws of physics and biology, contributing to stability and efficiency. Symmetry in real life is not just a mathematical concept but a universal language of balance and harmony.

## Conclusion

Symmetry, with its various types and applications, is not only a fascinating topic, but also an omnipresent concept weaving an intricate tapestry that marries art, nature, and mathematics. It’s an aesthetic quality that intuitively pleases the human eye and provides a sense of balance and order to the chaos of life.

It’s an idea that stretches far beyond our everyday life, creating a bridge between the macrocosm and the microcosm. From the perfect symmetry of a snowflake under a microscope to the grandeur of architectural designs like the Taj Mahal or the symmetry seen in iconic paintings such as the Mona Lisa, the presence of symmetry is as diverse as it is universal.

## Frequently Asked Questions on Symmetry

This section will address some of the most commonly asked questions about symmetry, helping to further clarify this fascinating topic. Whether you’re grappling with understanding the different types of symmetry, or you’re curious about its implications outside of mathematics, this section should clear up some of your queries.

### What is the line of symmetry?

A line of symmetry, also known as a mirror line or reflection line, is a line that divides a shape or figure into two identical halves. When a figure is folded along its line of symmetry, the two halves match exactly.

### What are some examples of symmetry in nature?

Symmetry is prevalent in nature. Examples include the bilateral symmetry of animals, the radial symmetry of flowers, the spherical symmetry of many fruits, and the intricate symmetry of snowflakes.

### What are the different types of symmetry?

There are several types of symmetry including reflective symmetry (mirror image), rotational symmetry (shape can be rotated and still look the same), translational symmetry (repeated patterns), glide symmetry (reflection followed by translation), and scale symmetry (symmetry at different scales or sizes).

### Why is symmetry important in mathematics?

Symmetry is important in mathematics because it helps simplify complex problems. By recognizing the symmetry in a problem, mathematicians can often reduce the complexity of calculations and visualize solutions more easily.

### Is symmetry only found in geometry?

No, symmetry is a broader concept found in many areas of mathematics and science, including algebra, calculus, and physics. In addition, symmetry plays a crucial role in art, architecture, and nature.